A collection of well-defined items is referred to as a set, in which there is a rule that can be used to determine if a given object belongs to the collection. A set can consist of any mathematical object, number, symbol, point in space, lines, other geometrical objects, variables, etc. Elements or members of the set are the objects that make up a set presentation. There are various ways to represent a set.

The elements of the set are marked by small letters a, b, c within brackets, and the set is denoted by capital letters A, B, C, D, etc.

• Roster Method: In the roaster form, the set is represented by keeping the elements within braces {.} which are separated by commas.

i.e. A set of vowels: V= {a, e, i, o, u}

A set of first six natural numbers: N= {1,2,3,4,5,6}

Note: The order in which the elements are written in a set makes no difference. Thus, {a, e, i, o, u} and {e,a, i,o, u} describe the same set. Additionally, the repetition of an element has no effect.

For example, the set {1, 2, 5, 7, 5, 4, 3, 2} is the same set as {1, 2, 3, 4, 5, 7}.

• Set-Builder/Rule Method: In the set-builder form, a set is represented by a characterizing property k(x) of their elements x. In this case, the set represented by {x:k(x) holds some conditions where we can read that a set of all x such that k(x) holds. Here, the symbol ‘:’ or ‘|’ is represented as ‘such that. 

i.e. the set E of all odd natural numbers can be written as; 

E={x:x is a natural number and x=(2n+1)}, for n ϵ N,

Set A={0, 1, 4, 9, 16…… }could be written as A={x: x=n2, n ϵ Z } 

Note: Some Important Symbols         

Different types of set presentation

Unit Sets or Singleton Sets

A unit set or singleton has only one element. Set A is equal to P where P is a prime number between 2 and 4, resulting in A being equal to {3}. It is one of the most common ways to represent a set.

Finite Sets

A finite set has a limited or finite number of elements. It is also called a non-empty finite set. An example of finite sets: Set B = P where P is an odd integer less than 10, for example, is A = {3,5,7,9}. 

Let’s assume the following example: Set B = {x: x is a day in a month}; here, set A would have 30 or 31 entries and which denotes that it is a finite set.

Infinite Sets

An infinite set contains an unlimited number of items. To put it another way, if a set was not finite, it would be endless. Let’s assume an example, B = {x: x is a real number}, and real numbers are infinite. As a result, A belongs to an infinite set. Let’s look at another case. Set C = {z: z is the coordinate of a  straight line}; is the coordinate of a point on a straight line; a straight line has an infinite number of points.

Empty Set

An empty set, also known as a null set that has no elements. The symbol’∅’ is used to represent an empty set. It’s pronounced ‘phi.’ Set X = {} as an example.

Sets of Equals

When two sets contain the same items, they are equal sets. For instance, B = {4,5,3} and A = {3,4,5}. Sets B and A are equal in this case. A = B can be used to represent this.

Unequal Sets

Unequal sets are those that have at least a single different element. For insurance, P= {5,6,7} and Q = {1,2,3}. Sets X and Y are unequal sets in this case. X ≠ Y can be used to symbolize this.

Conclusion

We have read about sets, their functions, formulas, and various examples in this article. A set is defined as the collection of well-defined items in which there is a rule that can be used to determine if a given object belongs or does not belong to the collection. A set can consist of any mathematical object, number, symbol, point in space, lines, other geometrical objects, variables, etc. We also learnt about the various ways to represent a set. Set is a crucial concept of mathematics, and it correlates with other topics such as domain, function, and range system. The set questions are one of the most simple topics to solve if you understand the concept.